Challenges and Paradigms in Applied Robust Control Part 3 potx

0 Fault Tolerant Depth Control of the MARES AUV Bruno Ferreira, Aníbal Matos and Nuno Cruz INESC Porto and University of Porto Portugal 1. Introduction Control theory has been applied to several domains where practical considerations are relevant. Robotics is a notable example of this. In most cases, mobile robotic systems are governed such that their behavior obeys to a defined motion. However, during their operations, it is conceivable that faults could occur. Indeed, this assumption has to be made in order to predict a possible malfunction and to take an appropriate action according to the fault, improving the robustness and the reliability of the system. This work tackles the problem of fault detection, identification and automatic reconfiguration of an autonomous underwater vehicle (AUV). Although our emphasis will be directed to an AUV, the methods and the tools that are employed in this chapter can be easily extended to other engineering problems beyond robotics. In this work, we will consider the MARES (Modular Autonomous Robot for Environment Sampling) (Fig. 1) Cruz & Matos (2008); Matos & Cruz (2009), a small-sized (1.5 meters long), torpedo shaped AUV weighting 32 kg, able to move at constant velocities up to 2.5 m/s. Its four thrusters provide four degrees of freedom (DOF), namely surge, heave, pitch and yaw. One of its main particularities is the capability to dive independently of the forward motion. The vertical through-hull thrusters provide heave and pitch controllability, while the horizontal ones ensure the surge and the yaw DOFs. The heave and pitch DOFs make the vertical plane control redundant when the vehicle is moving with surge velocities different from zero. In other words, the vehicle remains controllable if only one of these two DOFs is available. Such characteristic will be explored along this chapter in which the control of the nonlinear dynamics of the AUV Ferreira, Matos, Cruz & Pinto (2010); Fossen (1994) constitutes a challenging problem. By taking advantage of the distribution of the actuators on the vehicle, it is possible to decouple the horizontal and the vertical motion. A common approach in such systems is to consider reduced models in order to simplify the analysis and the derivation of the control law (see Ferreira, Matos, Cruz & Pinto (2010); Teixeira et al. (2010) or Fossen (1994), for example). In general, for topedo-shaped vehicles, coupling effects due to composed motions (e.g., simultaneous sway and heave motions) are clearly smaller than the self effects of decomposed motion (e.g., effect of the heave motion on the heave dynamics) and can therefore be considered disturbances in the reduced model in which they are not included. Thus, a reduced model will be considered to deal with the vertical motion taking surge, heave and pitch rate as state components. In order to make the detection and identification of possible faults, we present a method based on process monitoring by estimating relevant state variables of the system. See Frank & Ding (1997) for an overview on several techniques andZhang & Jiang (2002) for an application to a particular linear system. Wu et al. (2000) have developed an algorithm based on the two-stage 3 2 Will-be-set-by-IN-TECH Fig. 1. MARES starting a typical mission in the ocean Kalman filter to estimate deviations from expected input actuation for a linear system. Their approach consists in estimating the loss of control effectiveness factors that are added as entries of the state estimate, while guaranteeing that the corresponding estimate covariance lies in a defined interval. By imposing boundaries on the corresponding eigenvalues, it is possible to avoid impetuous corrections or to be insensitive to measurements. Inspired on the work by Zhang & Jiang (2002) and Wu et al. (2000), the present paper describes the implementation of an augmented state extended Kalman filter (ASEKF) to estimate the effectiveness of the control commands, detect and identify the possible faults. The present work focuses on the vertical motion considering faults on the vertical thrusters. The method for accomodation of the faults consists in three main steps: fault detection, fault diagnosis and decision. Fault detection is responsible for creating a warning whenever an abrupt or an incipient fault happens, while fault diagnosis distinguishes and identifies the fault. In the presence of faults, a decision must be taken, adopting a suitable control law to stabilize the vertical motion. In the presence of faults in one of the vertical thrusters, the heave motion will no longer be controllable. Consequently, a control law derived for normal operation could be inadequate or even turn the feedback system unstable when such a fault occurs. An algorithm has to be developed in order to make the behavior of the robotic system tolerant to faults. Making use of the pitch angle controllability, we will derive two control laws to drive the vehicle to a depth reference, possibly time variant. To achieve so, we make use of the Lyapunov theory, adopting the backstepping method Khalil (2002). Nevertheless, the presence of biases in steady state shifts the error at equilibrium away from zero. Those biases are commonly induced by unmodeled, neglected effects or external disturbances whose values are hard to observe or to estimate. The introduction of an integral term, under some assumptions, would solve the problem allowing the error to converge to zero as time goes to infinity. Based on the conditional integrators, extended by Singh & Khalil (2005) to more general control framework beyond sliding mode control Seshagiri & Khalil (2005), we present a control law that makes it possible to achieve asymptotic regulation of the vehicle depth error 50 Challenges and Paradigms in Applied Robust Control Fault Tolerant Depth Control of the MARES AUV 3 when operating with only one thruster. The method considers an integral component in the control law derived in the backstepping first step. Beyond the derivation and the particularization of the mathematical tools used here, we demonstrate our approach by illustrating the work with real experiments, voluntarily inducing faults on the system, and analyze the behavior of the dynamic system under such conditions. The organization of the chapter is as follow: the section 2 describes the MARES AUV and presents the main mathematical models under consideration here, namely the kinematic and the dynamic models. In section 3, we introduce the relevant concepts and formulate the ASEKF from a reduced model of the vertical dynamics and describe how to identify the fault. In section 4, we derive the control law to drive the depth error to zero making use of the tools described above. Finally, in 5, our solution is demonstrated through real experiments. 2. MARES The MARES autonomous underwater vehicle was developed in 2006 at the Faculty of Engineering of the University of Porto (FEUP). Typical operations have been performed in the ocean and fresh water, collecting relevant data for surveys and environmental monitoring during several tens of missions to date. Its configuration was specially designed to dive vertically in the water column while its horizontal motion is controlled independently, resulting in truly decoupled vertical and horizontal motions. Such characteristic is particularly appreciated in missions where the operation area is restricted or precise positioning is required. Parallel to the missions to collect data, the MARES AUV has also been used as a testbed for intensive research being performed in several problems related to robotics, specially on localization and control. Thus, besides the typical applications, several missions have been conducted to test and to verify implemented algorithms. Before presenting our method to detect a possible fault and to control the vehicle under such situation, let us first introduce the kinematics and dynamics concepts and equations. We assume an inertial earth-fixed frame {I} = {x I , y I , z I }, where x I , y I , z I ∈ R 3 are orthonormal vectors (in the marine literature, they are often made coincident with north, east and down directions, respectively), and a body-fixed frame {B} = {x B , y B , z B }, where x B , y B , z B ∈ R 3 are orthonormal vectors frequently refered to as surge, sway and heave directions, respectively (see Fig. 2). The absolute position and orientation of the vehicle is expressed in the inertial frame {I} through the vector η c =[η 1 , η 2 ] T =[x,y, z, φ, θ,ψ] T , where η 2 =[φ, θ, ψ] T is the vector of euler angles with respect to x I , y I and z I , and [x, y, z] are the coordinate of the frame {B} expressed in {I}. The vehicle’s velocity, expressed in the body frame {B}, is given by ν c =[u, v, w, p, q, r] T , where p,q and r are the angular velocities along x B , y B and z B , respectively. The velocities in both referentials are related through the kinematic equation Fossen (1994) ˙ η c = J(η 2 )ν c , (1) where ˙ η c denotes the time derivative of η and J(η 2 )=block diag[J 1 , J 2 ] represents the rotation matrix, with J 1 , J 2 ∈ R 3×3 . Although this transformation is common in the literature to map vectors from a referential frame to another, it is not the only one. An alternative can be found in quaternions (see Zhang (1997), for an introduction and useful results), avoiding the singularity problems of the matrix J 2 . However, in this chapter, we will assume that the values of the angles that make the matrix J 2 singular (and J, consequently) are not reached. Moreover, since the water currents present in the ocean and in the rivers do not influence the development of the present work, they will not be considered for simplicity. 51 Fault Tolerant Depth Control of the MARES AUV 4 Will-be-set-by-IN-TECH Fig. 2. Body frame and inertial frame referentials As it is well known, rigid bodies moving in the three dimensional space are governed by nonlinear equations. For the particular case of underwater vehicles, such equations include the effect of the added mass, viscous damping, restoring and actuation forces and moments. Following the notation in Fossen (1994), the nonlinear second order, six dimensions equation is written as: M c ˙ ν c = −C c (ν c )ν c − D c (ν c )ν c − g c (η c )+τ c (2) where M c ∈ R 6×6 is the sum of the body inertia and added mass matrices, C c ∈ R 6×6 results from the sum of the Coriolis and centrifugal terms from body inertia and added mass, D c ∈ R 6×6 is the viscous damping matrix, g c ∈ R 6 is the vector of restoring forces and moments and τ c ∈ R 6 is the vector of actuation forces and moments. Such system is complex and the task of deriving control laws that ensure stability is not trivial, having led to order reduction in several works (see Ferreira, Matos, Cruz & IEEE (2010); Fossen (1994); Teixeira et al. (2010), for example). By taking advantage of the body shape symmetries and of the configuration of the actuators on the body, it is usual to decouple the complex motion in more elementary ones. However, this has consequences since some cross-coupling terms are eliminated but their influence is often small and can be neglected or considered disturbances. Such approach has been implemented in the MARES AUV and the corresponding performances were already demonstrated in Ferreira, Matos, Cruz & Pinto (2010). The current thruster configuration on the MARES makes it possible to decouple the motion into the vertical and the horizontal plane. Since the roll angle is stable (and φ ≈ 0) two reduced order models are extracted. See Ferreira, Matos, Cruz & Pinto (2010), for further details. 3. Fault detection and identification Under normal operation, the vertical thrusters of MARES provide the capability to control almost independently the pitch and heave degrees of freedom (DOF). In the same way, the horizontal thrusters make possible the control on the surge and yaw DOFs. As it will be exposed later, the vehicle remains controllable if one of the vertical thrusters fails. As an aside, note that the same is not verified if one the two horizontal thrusters fails since the surge and the yaw motion can no longer be decoupled. The derivation of the control laws is left to the next section. 52 Challenges and Paradigms in Applied Robust Control Fault Tolerant Depth Control of the MARES AUV 5 Several approaches for fault identification and detection have been proposed, commonly based on observation of the residuals Frank & Ding (1997) either by using state observers or by using accurate models. Our approach makes use of an extended Kalman filter to detect the faults, indirectly exploiting the residuals to estimate actuation bias variables additionally incorporated in the state. A similar approach was already carried out by Zhang & Jiang (2002) where they used a two stage Kalman filter to identify the faults on actuation of a linear system. The method presented here makes it possible not only to detect faults but also to identify the faulty actuator. By taking advantage of the cyclic predictions and corrections, the main idea behind our approach is to estimate the biases on the actuation (or deviation from the nominal value, commonly referred to as loss of control effectiveness factor in Wu et al. (2000); Zhang & Jiang (2002) whose values should theoretically equal zero when no fault is occuring. 3.1 Vertical plane dynamics The implementation of the extended Kalman filter assumes the use of a reasonably accurate dynamics model that recreates mathematically the behavior of the system for the prediction step. As it was pointed out in the previous section, the use of the complete model of the vehicle dynamics is complex and computationally expensive. Thus, from 2, we derive the reduced model for the vertical motion, considering that cross-terms are negligible: M ˙ ν = −C(ν)ν −D(ν)ν −g (η 2 )+P f f t (t) (3) where ν =[u, w, q] T and M = ⎡ ⎢ ⎣ m − X ˙ u 0 −X ˙ q 0 m − Z ˙ w −Z ˙ q −M ˙ u −M ˙ w −M ˙ q ⎤ ⎥ ⎦ , C (ν)= ⎡ ⎢ ⎣ 0 −mq −Z ˙ w w − Z ˙ q q −mq 0 X ˙ u u + X ˙ q q Z ˙ w w + Z ˙ q q −X ˙ u u − X ˙ q q 0 ⎤ ⎥ ⎦ , D (ν)=− ⎡ ⎢ ⎣ X |u|u |u| 0 X |q|q |q| 0 Z |w|w |w| Z |q|q |q| M |u|u |u| M |w|w |w| M |q|q |q| ⎤ ⎥ ⎦ , g (η 2 )=− ⎡ ⎢ ⎣ (W − B) sin θ (B −W) cos θ −z CB B sin θ ⎤ ⎥ ⎦ , P f = ⎡ ⎢ ⎣ 11 0 0 00 1 1 00x ts x tb ⎤ ⎥ ⎦ , f t (t)= ⎡ ⎢ ⎢ ⎣ f p (t) f r (t) f s (t) f b (t) ⎤ ⎥ ⎥ ⎦ . The authors recommend Fossen (1994); Triantafyllou & Hover (2002) for details about the parameters above and their derivation, and Hoerner (1993); White (2008) for further details. Notice that f t (t) is the vector of forces applied by the thrusters that are generated according to a given control law, and f p , f r , f s and f b are scalars that represents the force applied by the port, starboard, stern and bow thrusters, respectively. We assume that such forces can be directly measured during operation. The inclusion of the surge velocity is required in this reduced order model due to the nonnegligible influence it has on the vertical plane dynamics. The parameters used in the reduced model are listed in the table 1. 3.2 Augmented state extended Kalman filter formulation Our final goal in this section is to detect and to identify a fault occuring on one of the vertical thrusters. To this end, one aims to quantify the loss of control effectiveness of the referred actuators: The effective force applied by the vertical thrusters may differ from the commanded one. We will consider that f s and f b are the commanded forces, which may not correspond 53 Fault Tolerant Depth Control of the MARES AUV 6 Will-be-set-by-IN-TECH Parameter Value units Description m 3.20 ·10 1 kg Vehicle’s mass W 3.14 ·10 2 N Vehicle’s weight B 3.16 ·10 2 N Vehicle’s bouyancy z CB −4.40 ·10 −3 mz B of CB w.r.t CG X ˙ u −1.74 ·10 0 kg Added mass longitudinal term X ˙ q −3.05 ·10 −2 kg · m Added mass cross-term Z ˙ w −4.12 ·10 1 kg Added mass heave term Z ˙ q −1.23 ·10 −1 kg · m Added mass cross-term M ˙ u −3.05 ·10 −2 kg · m Added mass cross-term M ˙ w −1.23 ·10 −1 kg · m Added mass cross-term M ˙ q −6.07 ·10 0 kg · m 2 Added mass pitch term X |u|u −1.04 ·10 1 kg · m −1 Drag longitudinal term X |q|q 4.84 ·10 −2 kg · m Drag cross term Z |w|w −1.16 ·10 2 kg · m −1 Drag heave term Z |q|q −5.95 ·10 0 kg · m Drag cross-term M |u|u −2.11 ·10 −1 kg Drag cross-term M |w|w −8.26 ·10 0 kg Drag cross-term M |q|q −1.56 ·10 1 kg · m 2 Drag pitch term x ts −3.21 ·10 −1 mx B of stern vertical thruster w.r.t CG x tb 5.34 ·10 −1 mx B of bow vertical thruster w.r.t CG Table 1. Reduced model terms to the effective applied force. Like in many other problems in robotics, it is often difficult or even impossible to measure such forces. Measuring relative or absolute motion variables then becomes an alternative and the choice of the state to be observed is directly influenced by the variables that can be actually measured. Therefore, we propose the following model for the fault free ideal system: ˙ x =  ˙x 1 ˙x 2  = ⎡ ⎢ ⎢ ⎣ ˙ z ˙ θ ˙ w ˙ q ⎤ ⎥ ⎥ ⎦ = A l (x)x + f u (x,u)+ f (x,u, f v )+w x (t) (4) y = h(x)+v n (t), where w x ∈ R 4 is a zero-mean Gaussian noise vector with autocorrelation matrix Q w (t), x 1 =[z, θ] T , x 2 =[w, q] T , A l and f u (·) are easily derived from the kinematics model in 1 as A l (x)= ⎡ ⎢ ⎣ 0 0 cos θ 0 00 0 1 00 0 0 00 0 0 ⎤ ⎥ ⎦ , f u (x,u)= ⎡ ⎢ ⎣ −u sin θ 0 0 0 ⎤ ⎥ ⎦ , 54 Challenges and Paradigms in Applied Robust Control Fault Tolerant Depth Control of the MARES AUV 7 assuming φ = 0, and f (·) results from the dynamics model 3 as f (x,u, f v )= ⎡ ⎢ ⎣ 0 0 ˙ w ˙ q ⎤ ⎥ ⎦ = S ˙ ν = SM −1  −C(ν)ν −D(ν)ν −g(x 2 )+P v f v (t)  , S = ⎡ ⎢ ⎣ 000 000 010 001 ⎤ ⎥ ⎦ , P v = ⎡ ⎣ 00 11 x ts x tb ⎤ ⎦ , f v =  f s (t) f b (t)  . Regarding the output y of 4, the dimension of the function h (x) depends on the measurements and consequently on the on-board sensors. Here, we will assume we are able to observe the depth z, the pitch angle θ and the pitch rate q. Thus it results h (x)=C h x, C h = ⎡ ⎣ 1000 0100 0001 ⎤ ⎦ . The vector v n ∈ R 3 is the output noise, assumed to be zero-mean, Gaussian noise with autocorrelation matrix R v . Notice that x 2 is the vector containing the last two entries of the velocity vector ν, i.e., ν = [ u, x 2 ]. For simplicity of notation, in the expressions above we wrote ν instead of [u, x 2 ]. Recall that we assumed that u is a measured variable, or at least, it can be accurately estimated. Indeed, it could be included in the state in 4 but the complexity of this latter would increase without advantages in the approach. In order to model the possible loss of control effectiveness, let us define γ =[γ s , γ b ] T as the vector of loss of control effectiveness factors, adopting the same notation as in Wu et al. (2000). Introducing these multiplicative factors in 4, the augmented state model results in ˙ x = A l (x)x + f u (x,u)+ f (x,u, f v )+E(f v )γ + w x (t) ˙ γ = w γ (t) (5) y = C h x + v n (t), where w γ ∈ R 2 is a zero-mean, Gaussian noise vector with autocorrelation matrix Q γ , uncorrelated with w x , and E (f v )=SM −1 P v diag(f v ). As it can be seen in 5, γ is assumed to be driven only by the noise w γ . This comes from the fact that, in real scenarios, it is impossible to predict how the fault and, consequently, how γ evolve. In such situation, the most appropriate is to model the evolution with a noise vector w γ with a sufficiently large autocorrelation (see Wu et al. (2000)), whose entries can play an important role in the design of the augmented state estimator, as it will be seen later on. Making s =[x T , γ T ] T , we rewrite 5 on the form ˙ s = A s (s)s + f us (s,u)+ f s (s,u, f v )+E s (f v )s + w s (t) (6) y = C s s + v n (t). 55 Fault Tolerant Depth Control of the MARES AUV 8 Will-be-set-by-IN-TECH where A s =  A l 0 4×2 0 2×4 0 2×2  , f us =  f u 0 2×1  , f s =  f 0 2×1  , E s =  0 4×4 E 0 2×4 0 2×2  , w s =  w x w γ  C s =  C h 0 3×2  . The discrete time representation of 6 follows s k+1 = A sk (s k )s k + f usk (s k , u k )+ f sk (s k , u k , f v k )+E sk (f v k )s k + w sk (7) y k+1 = C s s k+1 + v k+1 , where β k represents the discrete time equivalent vector, or matrix, β at time t k . We assume that the process noise w sk and the output noise v k are uncorrelated, i.e., E {w sk v T k } = 0. The autocorrelation of the process noise and of the output are respectively given by E {w sk w T sk } = Q k =  Q x k 0 0 Q γ k  , E {v k v T k } = R k . (8) The formulation of a Kalman filter assumes the use of a model of the process which is a mathematical representation of the dynamics. However, the mathematical translation of the dynamics of a given system may be inaccurate or may not describe entirely its behavior. This is the case in hydrodynamics, where the models are complex, difficult to extract. Moreover, there is no complete theory that allows for determining an accurate model and calculations of parameters mostly rely on empirical or semi-empirical formulas. Hence, we define ˆ β as the estimate of the generic vector, or matrix, β. The augmented state extended Kalman filter formulation follows now directly from Gelb (1974). During the prediction stage, the state estimate and the covariance matrix evolve according to ˆ s k+1|k = ˆ A ˆ sk ( ˆ s k ) ˆ s k + ˆ f usk ( ˆ s k , ˆ u k )+ ˆ f sk ( ˆ s k , ˆ u k , f v k )+ ˆ E ˆ sk (f v k ) ˆ s k (9) P k+1|k = F k P k F T k + Q k , (10) where F k stands for the Jacobian of ˙ s evaluated at ˆ s k : F k = ∂ ˙ s ∂s |s= ˆ s k . The so-called Kalman gain and the updates of the estimate and of the covariance matrix are respectively given by K k+1 = P k+1|k C T s (C s P k+1|k C T s + R k ) −1 (11) ˆ s k+1|k+1 = s k+1|k + K k+1 (y k+1 −C s s k+1|k ) (12) P k+1|k+1 =(I −K k+1 C s )P k+1|k . (13) From the state estimate, it is now possible to extract the vector γ k , whose entries constitute the base to determine whether a fault has occured or not. As it was stated earlier, the autocorrelation matrix Q γ k can play a significant role to avoid divergence or guarantee faster convergence of the estimate of the loss of control effectiveness 56 Challenges and Paradigms in Applied Robust Control Fault Tolerant Depth Control of the MARES AUV 9 factors. For the sake of clarity, from 10 and 13 we can decompose P k as P k =  P x k P γx k P xγ k P γ k  Thus, from 10 and 13, we can conclude P γ k +1|k = P γ k |k + Q γ k P γ k +1|k+1 ≤ P γ k +1|k where we used the fact that P k+1|k > 0. Hence, the autocorrelation matrix Q γ k can be set such that P γ lies in an interval, preventing state corrections to be excessive, when P γ is too large, or to be insufficient with slow convergence, when P γ is too small. Taking the eigenvalues λ γ of P γ as measures, we propose the following function Q γ =  diag (q γ , q γ ) , if max(λ γ 1 , λ γ 2 ) < λ max 0 , if max(λ γ 1 , λ γ 2 ) ≥ λ max (14) where q γ is the autocorrelation of γ i , i = 1, 2 and λ max is a preset maximum constant. 3.3 Fault identification The loss control effectiveness factors provide an estimate of the performances of the actuators. Ideally, a fault would be identified whenever the absolute value of one of the factors would rise above a preset threshold. However, model uncertainties will be directly reflected in these factors. Even in normal operation, with the actuators working perfectly, the loss of control effectiveness factors may diverge from zero, reflecting, for example, the effect of a damping force greater than the modeled. As these errors are frequently commited on the overall model, their effects are verified on all actuators either by increasing or decreasing the loss of control effectiveness factors. Hence, for the present case, a reasonable measure of the malfunction of one of the thrusters is given by the difference of the corresponding loss of control effectiveness factor estimate. On the other hand, taking a decision about the malfunction of a given thruster should also be based on the confidence of the factor estimate, which can be indirectly taken from the eigenvalues λ γ of P γ k , avoiding taking decisions on transient state, while considerable corrections on the state are being performed. Thus, we propose the following measure for fault detection: δ = | γ s −γ b | f λ (λ γ 1 + λ γ 2 ) . (15) where f λ is a monotically increasing function of its argument. Whenever δ is greater than a preset threshold, a fault is detected and the identification is made according to the greater λ, i.e., if γ s > γ b then the stern thruster is faulty and vice-versa. 4. Control of MARES In the presence of a faulty vertical thruster, the reconfiguration of the actuation is required. Otherwise, keeping the same actuation will likely lead to instability or to other pratical problems such as thruster dammage or large battery consumption, for example. Therefore, the control law for normal operation could be inadequate and another control law must take 57 Fault Tolerant Depth Control of the MARES AUV 10 Will-be-set-by-IN-TECH Fig. 3. Operation of the fault overall fault detection and recovery algorithm over. In this section, we first start by deriving such controller and present the main concepts behind the derivation of the control for normal operation in order to make the result section clear. 4.1 Control under fault We consider now the scenario in which only one of the vertical thrusters is available to control the motion of the vehicle. Under such situation, the heave DOF is no longer controllable but the depth is still controllable by manipulating pitch. Based on the Lyapunov theory we will derive a controller that makes it possible to control the vehicle’s depth, while assuming that the absolute value of the surge velocity is sufficiently large to compensate the vehicle’s flotation. The derivation of the controller employs the well know backstepping method as well as conditional integrators to achieve asymptotic regulation. As the final goal in this section is to control the vehicle depth, we will assume that roll angle is null (φ = 0), resulting: ˙ z = −u sin θ + w cos θ (16) Let us introduce the error variable e z = z − z d , which we want to drive to zero, and the quadratic Lyapunov function: V 1 = 1 2 e 2 z , (17) whose time derivative results ˙ V 1 = e z ˙ e z = e z (−u sin θ + w cos θ − ˙ z d ). (18) Although u,θ and z are measured by sensors or estimated, it is hard to accurately compute w due to model uncertainties and measurement noise. Thus we will assume that it constitutes a disturbance acting on the system, shifting the equilibrium point e z = 0 to an uncertain value. Throughout the following developments, we will consider that the surge velocity is maintained constant in order to simplify our approach. Indeed, in most missions the surge velocity is intended to be constant along the trajectory. Moreover, the limited actuation on the vertical thruster makes the pitch angular velocity to lie in a bounded interval. Hence, from the vertical dynamics, we can assume that there exists an upper bound on the absolute value of w ∈ [−w max , w max ]. 58 Challenges and Paradigms in Applied Robust Control [...]... nonlinear systems using conditional integrators, International Journal of Robust and Nonlinear Control 15(8): 33 9 36 2 930 VR Teixeira, F C., Aguiar, A P & Pascoal, A (2010) Nonlinear adaptive control of an underwater towed vehicle, Ocean Engineering 37 ( 13) : 11 93 1220 Triantafyllou, M S & Hover, F S (2002) Maneuvering and control of marine vehicles, Technical report, Massachussets Institute of Technology... trajectory reaches the sliding manifold the corresponding sliding variable σ (x) is equal to zero Since σ (x) appears 76 4 Challenges and Paradigms in Applied Robust Control Robust Control u in the denominator of the overall control gain k = σ this variable is drastically increased In practice that means that the discontinuous control law acts directly with its maximum but finite control input if the system... still able to satisfy all current ISC design requirements 2 Sliding mode control and second order sliding mode control Sliding mode control theory has attracted great interest among scientists and control engineers within the last decades The resulting control laws can be applied but are not restricted to affine nonlinear single input single output (SISO) systems ˙ x(t) = f (x(t)) + g (x(t)) u(t) +... drastically increased (Ehsani et al (2010)) an ongoing rise on development cost is inevitable However, with this effect cars may become 74 2 Challenges and Paradigms in Applied Robust Control Robust Control unaffordable to many customers in the near future Thus, novel control design strategies have to be introduced such that today’s and future calibration work is minimized y y1 y2 y y y3 y1 y y u y2 y y y y3... uncertainties and external disturbances where Φ, Γm , Γ M ∈ R + Thus, the robustness properties are considered to be similar to those of a closed-loop system with first order sliding mode control law 78 6 Challenges and Paradigms in Applied Robust Control Robust Control For a better general understanding the reduction of the chattering effects can be related to the additional integrator within the... disturbances induce undesired effects on the controllers However, the following figures show the robustness of our approach The variables shown in the graphs have the following units correspondence: Depth is expressed in meters, the pitch angle is expressed in radians while the surge velocity is expressed in meters per second 64 16 Challenges and Paradigms in Applied Robust Control Will-be-set-by -IN- TECH (a) Control. .. angle (c) Surge velocity Fig 10 MARES controlling depth with bow and horizontal thrusters only (zd = 0.6 m) 70 22 Challenges and Paradigms in Applied Robust Control Will-be-set-by -IN- TECH founds in practical operations Moreover, the commanded forces are affected by delays and thruster model is subject to uncertainties, which certainly in uence the behavior On the other hand, the Fig 11(a) shows that the... control design (Edwards & Spurgeon (1998)) Hence, the total number of required operating points can be considerably reduced thus leading to less calibration efforts (see Figure 3) Moreover, sliding mode control shows good robustness properties against a wide class of model uncertainties and external disturbances including environmental in uences, aging and tolerance effects (Hung et al (19 93) ; Utkin... the sliding manifold is affected Due to that high gain effect the robustness properties of the sliding mode control system are similar to a closed-loop system with high-gain control law (Khalil (1996)) On the contrary to this class of nonlinear controllers the corresponding sliding mode control input doesn’t suffer from unrealistic large control efforts Instead it is well known that this control input... σ (x) > 0 (3) As soon as the system trajectory reaches the sliding manifold σ (x) = 0 the control input usmc shows a switching effect with in nite frequency Of course, this in nite fast switching effect cannot occur in practical applications since each actuator has a limited bandwith and the corresponding control laws are calculated with finite sampling rates Thus, the intended ideal sliding motion . error 50 Challenges and Paradigms in Applied Robust Control Fault Tolerant Depth Control of the MARES AUV 3 when operating with only one thruster. The method considers an integral component in the control. monotically increasing along the operation due to the reduced actuation, and consequently poor observability. In 62 Challenges and Paradigms in Applied Robust Control Fault Tolerant Depth Control. the loss of control effectiveness 56 Challenges and Paradigms in Applied Robust Control Fault Tolerant Depth Control of the MARES AUV 9 factors. For the sake of clarity, from 10 and 13 we can decompose